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A reduction principle for nonautonomous systems in infinite-dimensional spaces. (English) Zbl 0601.35018

A generalization of the center manifold theorem is given which applies to semilinear parabolic and semilinear damped hyperbolic equations as well as to semilinear elliptic equations in cylindrical domains. In a Banach space the equation \[ (1)\quad \dot x-Lx=F(t,\lambda,x) \] is considered, where \(f(t,\lambda_ 0,0)=(\partial /\partial x)f(t,\lambda_ 0,0)=0.\) If the spectral part of L on the imaginary axis is finite-dimensional, the set of all small bounded solutions can be determined by reducing the problem to an ordinary differential equation \[ (2)\quad \dot x_ 1- Lx_ 1=f_ 1(t,\lambda,x_ 1+h(t,\lambda,x_ 1)). \] It is shown that symmetries of (1) with respect to x as well as (quasi)-periodicity with respect to t are inherited by (2). Moreover, asymptotic autonomy transfers from (1) to (2). Applications to solitary water waves under external forcing are given.

MSC:

35G30 Boundary value problems for nonlinear higher-order PDEs
35B32 Bifurcations in context of PDEs
34C45 Invariant manifolds for ordinary differential equations
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